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Secp256k1-types

Secp256k1-field

Fixtype of the elements of the secp256k1 field.

These are natural numbers below the prime p.

Definitions and Theorems

Function: secp256k1-fieldp

(defun secp256k1-fieldp (x)
  (declare (xargs :guard t))
  (integer-range-p 0 (secp256k1-field-prime)
                   x))

Theorem: booleanp-of-secp256k1-fieldp

(defthm booleanp-of-secp256k1-fieldp
  (b* ((yes/no (secp256k1-fieldp x)))
    (booleanp yes/no))
  :rule-classes :rewrite)

Theorem: natp-and-below-prime-when-secp256k1-fieldp

(defthm natp-and-below-prime-when-secp256k1-fieldp
 (implies
  (secp256k1-fieldp x)
  (and
   (natp x)
   (<
    x
    115792089237316195423570985008687907853269984665640564039457584007908834671663)))
 :rule-classes :tau-system)

Function: secp256k1-field-fix

(defun secp256k1-field-fix (x)
  (declare (xargs :guard (secp256k1-fieldp x)))
  (mbe :logic (if (secp256k1-fieldp x) x 0)
       :exec x))

Theorem: secp256k1-fieldp-of-secp256k1-field-fix

(defthm secp256k1-fieldp-of-secp256k1-field-fix
  (b* ((fixed-x (secp256k1-field-fix x)))
    (secp256k1-fieldp fixed-x))
  :rule-classes :rewrite)

Theorem: secp256k1-field-fix-when-secp256k1-fieldp

(defthm secp256k1-field-fix-when-secp256k1-fieldp
  (implies (secp256k1-fieldp x)
           (equal (secp256k1-field-fix x) x)))

Function: secp256k1-field-equiv$inline

(defun secp256k1-field-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (secp256k1-fieldp acl2::x)
                              (secp256k1-fieldp acl2::y))))
  (equal (secp256k1-field-fix acl2::x)
         (secp256k1-field-fix acl2::y)))

Theorem: secp256k1-field-equiv-is-an-equivalence

(defthm secp256k1-field-equiv-is-an-equivalence
  (and (booleanp (secp256k1-field-equiv x y))
       (secp256k1-field-equiv x x)
       (implies (secp256k1-field-equiv x y)
                (secp256k1-field-equiv y x))
       (implies (and (secp256k1-field-equiv x y)
                     (secp256k1-field-equiv y z))
                (secp256k1-field-equiv x z)))
  :rule-classes (:equivalence))

Theorem: secp256k1-field-equiv-implies-equal-secp256k1-field-fix-1

(defthm secp256k1-field-equiv-implies-equal-secp256k1-field-fix-1
  (implies (secp256k1-field-equiv acl2::x x-equiv)
           (equal (secp256k1-field-fix acl2::x)
                  (secp256k1-field-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: secp256k1-field-fix-under-secp256k1-field-equiv

(defthm secp256k1-field-fix-under-secp256k1-field-equiv
  (secp256k1-field-equiv (secp256k1-field-fix acl2::x)
                         acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-secp256k1-field-fix-1-forward-to-secp256k1-field-equiv

(defthm
    equal-of-secp256k1-field-fix-1-forward-to-secp256k1-field-equiv
  (implies (equal (secp256k1-field-fix acl2::x)
                  acl2::y)
           (secp256k1-field-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-secp256k1-field-fix-2-forward-to-secp256k1-field-equiv

(defthm
    equal-of-secp256k1-field-fix-2-forward-to-secp256k1-field-equiv
  (implies (equal acl2::x (secp256k1-field-fix acl2::y))
           (secp256k1-field-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: secp256k1-field-equiv-of-secp256k1-field-fix-1-forward

(defthm secp256k1-field-equiv-of-secp256k1-field-fix-1-forward
  (implies (secp256k1-field-equiv (secp256k1-field-fix acl2::x)
                                  acl2::y)
           (secp256k1-field-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: secp256k1-field-equiv-of-secp256k1-field-fix-2-forward

(defthm secp256k1-field-equiv-of-secp256k1-field-fix-2-forward
  (implies
       (secp256k1-field-equiv acl2::x (secp256k1-field-fix acl2::y))
       (secp256k1-field-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Subtopics

Secp256k1-fieldp: Recognizer for secp256k1-field.
Secp256k1-field-fix: Fixer for secp256k1-field.